The proof of the Goldbach's Conjecture is one of the biggest still unsolved problems regarding prime numbers. Originally expressed in 1742 by the mathematician Christian Goldbach, from whom the conjecture takes its name, it was rephrased by Euler in the form in which we know it today:

**Every even number greater than 2 can be expressed as a sum of two prime numbers.**

In spite of the empirical evidence and the simplicity of its statement, the conjecture is resisting to every attempts of proof since almost three centuries. Several mathematicians have proved some weaker versions of the conjecture.

The statement phrased by Christian Goldbach is a "conjecture", so, as a matter of principle, it is a hypothesis. This means that it can be:

- True, that is all even numbers greater than 2 can be expressed as a sum of two prime numbers;
- False, that is at least one even number greater than two exists, which cannot be written as a sum of two prime numbers.

Currently, there are several attempts to prove the Goldbach's conjecture, which are complete, in the sense that they come to the conclusion, but contain a series of problems, so they have not been considered valid by the international community. We instead propose some proof strategies, which are still far from being complete, but already contain interesting and unrefuted (so far) intermediate results. We hope this material will be a useful starting point for who, like us, has started looking for a proof...

- True, that is all even numbers greater than 2 can be expressed as a sum of two prime numbers;
- False, that is at least one even number greater than two exists, which cannot be written as a sum of two prime numbers.

Goldbach's conjecture is put into the field of Number theory, the branch of Mathematics which studies the properties of integer numbers. In order to understand the proofs of the results similar to the conjecture, and very likely also to prove the conjecture itself, a sound knowledge of number theory is required. But this kind of knowledge rarely is part of a mathematician's curriculum...

Dashed line theory is a new mathematical theory which studies the connection between the sequence of natural numbers and their divisibility relationship. Typical problems are the computation of the n-th natural number divisible by at least one of k fixed numbers, or not divisible by any of them. Due to this nature, the theory is suited for studying prime numbers with a constructive approach, inspired to the sieve of Eratosthenes...

# Latest posts

Given an even number N > 2, we'll call Goldbach pairs for N the pairs of prime numbers the sum of which is N. Goldbach's Conjecture states that, whatever N, the number of these pairs is at least 1. In this article we'll reason in the opposite direction: we'll find an overestimate of the number of Goldbach pairs, that is, we'll establish how many they can be at most. We'll do this using the sieve theory developed so far, in particular Selberg's sieve.

This article concludes the "technical" part of the proof of Selberg's sieve and lays the foundation for the applications that we'll see later. The parts of the proof that we have seen in the previous articles will lead us, in a few steps, to the statement from which we started, Theorem C.4, which however is only a simplified version of the theorem. In this article we'll also state its ordinary form and we'll see that its proof differs only in a few points from that of the simplified version.

In number theory, to understand some proofs, a certain familiarity with the concept of "numerical series" is necessary. In this article, after briefly introducing this concept, we'll focus on some series used in number theory, which are not always studied at school or university; we'll see in particular some techniques which are useful in practice to operate with such series. This article does not claim to treat numerical series in a rigorous nor exhaustive way, but it can be seen as a complementary study for those who have already studied series, and as a starting point for further study for…

In this article we'll discuss arithmetic functions, that are functions defined on positive integers; the value of such functions, however, may not always be an integer. Some arithmetic functions are well-known and recurrent in number theory, and we have introduced them in our articles where appropriate. Here we'll summarize their definitions and then we'll see some of their properties that can potentially be of common use, also with regard to the techniques used in some proofs.

We have seen that, thanks to Selberg's sieve, an estimate of a sieve function can be calculated; this estimate is made up of a term T which represents the approximation of the function, and a term R which represents the error. In the previous article we calculated the λ parameters that make the T part as low as possible. Now all that remains is to replace these parameter values in the R error function; this will be the purpose of the present article.